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Theorem suplocexprlemlub 7525
Description: Lemma for suplocexpr 7526. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocexpr.ub (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
suplocexpr.loc (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
suplocexpr.b 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
Assertion
Ref Expression
suplocexprlemlub (𝜑 → (𝑦<P 𝐵 → ∃𝑧𝐴 𝑦<P 𝑧))
Distinct variable groups:   𝑦,𝐴,𝑧   𝑥,𝐴,𝑦   𝑧,𝐵   𝜑,𝑦,𝑧   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑤,𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦,𝑤,𝑢)

Proof of Theorem suplocexprlemlub
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . 4 ((𝜑𝑦<P 𝐵) → 𝑦<P 𝐵)
2 ltrelpr 7306 . . . . . . . 8 <P ⊆ (P × P)
32brel 4586 . . . . . . 7 (𝑦<P 𝐵 → (𝑦P𝐵P))
43simpld 111 . . . . . 6 (𝑦<P 𝐵𝑦P)
54adantl 275 . . . . 5 ((𝜑𝑦<P 𝐵) → 𝑦P)
63simprd 113 . . . . . 6 (𝑦<P 𝐵𝐵P)
76adantl 275 . . . . 5 ((𝜑𝑦<P 𝐵) → 𝐵P)
8 ltdfpr 7307 . . . . 5 ((𝑦P𝐵P) → (𝑦<P 𝐵 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵))))
95, 7, 8syl2anc 408 . . . 4 ((𝜑𝑦<P 𝐵) → (𝑦<P 𝐵 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵))))
101, 9mpbid 146 . . 3 ((𝜑𝑦<P 𝐵) → ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))
11 simprrr 529 . . . . . 6 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → 𝑠 ∈ (1st𝐵))
12 suplocexpr.b . . . . . . . . . 10 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
1312fveq2i 5417 . . . . . . . . 9 (1st𝐵) = (1st ‘⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩)
14 npex 7274 . . . . . . . . . . . . 13 P ∈ V
1514a1i 9 . . . . . . . . . . . 12 (𝜑P ∈ V)
16 suplocexpr.m . . . . . . . . . . . . 13 (𝜑 → ∃𝑥 𝑥𝐴)
17 suplocexpr.ub . . . . . . . . . . . . 13 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
18 suplocexpr.loc . . . . . . . . . . . . 13 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
1916, 17, 18suplocexprlemss 7516 . . . . . . . . . . . 12 (𝜑𝐴P)
2015, 19ssexd 4063 . . . . . . . . . . 11 (𝜑𝐴 ∈ V)
21 fo1st 6048 . . . . . . . . . . . . 13 1st :V–onto→V
22 fofun 5341 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
2321, 22ax-mp 5 . . . . . . . . . . . 12 Fun 1st
24 funimaexg 5202 . . . . . . . . . . . 12 ((Fun 1st𝐴 ∈ V) → (1st𝐴) ∈ V)
2523, 24mpan 420 . . . . . . . . . . 11 (𝐴 ∈ V → (1st𝐴) ∈ V)
26 uniexg 4356 . . . . . . . . . . 11 ((1st𝐴) ∈ V → (1st𝐴) ∈ V)
2720, 25, 263syl 17 . . . . . . . . . 10 (𝜑 (1st𝐴) ∈ V)
28 nqex 7164 . . . . . . . . . . 11 Q ∈ V
2928rabex 4067 . . . . . . . . . 10 {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ∈ V
30 op1stg 6041 . . . . . . . . . 10 (( (1st𝐴) ∈ V ∧ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ∈ V) → (1st ‘⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩) = (1st𝐴))
3127, 29, 30sylancl 409 . . . . . . . . 9 (𝜑 → (1st ‘⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩) = (1st𝐴))
3213, 31syl5eq 2182 . . . . . . . 8 (𝜑 → (1st𝐵) = (1st𝐴))
3332eleq2d 2207 . . . . . . 7 (𝜑 → (𝑠 ∈ (1st𝐵) ↔ 𝑠 (1st𝐴)))
3433ad2antrr 479 . . . . . 6 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → (𝑠 ∈ (1st𝐵) ↔ 𝑠 (1st𝐴)))
3511, 34mpbid 146 . . . . 5 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → 𝑠 (1st𝐴))
36 suplocexprlemell 7514 . . . . 5 (𝑠 (1st𝐴) ↔ ∃𝑧𝐴 𝑠 ∈ (1st𝑧))
3735, 36sylib 121 . . . 4 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → ∃𝑧𝐴 𝑠 ∈ (1st𝑧))
38 simprl 520 . . . . . . . . 9 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → 𝑠Q)
3938ad2antrr 479 . . . . . . . 8 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑠Q)
40 simprrl 528 . . . . . . . . 9 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → 𝑠 ∈ (2nd𝑦))
4140ad2antrr 479 . . . . . . . 8 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑠 ∈ (2nd𝑦))
42 simpr 109 . . . . . . . 8 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑠 ∈ (1st𝑧))
43 rspe 2479 . . . . . . . 8 ((𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝑧))) → ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝑧)))
4439, 41, 42, 43syl12anc 1214 . . . . . . 7 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝑧)))
454ad4antlr 486 . . . . . . . 8 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑦P)
4619ad4antr 485 . . . . . . . . 9 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝐴P)
47 simplr 519 . . . . . . . . 9 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑧𝐴)
4846, 47sseldd 3093 . . . . . . . 8 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑧P)
49 ltdfpr 7307 . . . . . . . 8 ((𝑦P𝑧P) → (𝑦<P 𝑧 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝑧))))
5045, 48, 49syl2anc 408 . . . . . . 7 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → (𝑦<P 𝑧 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝑧))))
5144, 50mpbird 166 . . . . . 6 (((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) ∧ 𝑠 ∈ (1st𝑧)) → 𝑦<P 𝑧)
5251ex 114 . . . . 5 ((((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) ∧ 𝑧𝐴) → (𝑠 ∈ (1st𝑧) → 𝑦<P 𝑧))
5352reximdva 2532 . . . 4 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → (∃𝑧𝐴 𝑠 ∈ (1st𝑧) → ∃𝑧𝐴 𝑦<P 𝑧))
5437, 53mpd 13 . . 3 (((𝜑𝑦<P 𝐵) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝑦) ∧ 𝑠 ∈ (1st𝐵)))) → ∃𝑧𝐴 𝑦<P 𝑧)
5510, 54rexlimddv 2552 . 2 ((𝜑𝑦<P 𝐵) → ∃𝑧𝐴 𝑦<P 𝑧)
5655ex 114 1 (𝜑 → (𝑦<P 𝐵 → ∃𝑧𝐴 𝑦<P 𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 697   = wceq 1331  wex 1468  wcel 1480  wral 2414  wrex 2415  {crab 2418  Vcvv 2681  wss 3066  cop 3525   cuni 3731   cint 3766   class class class wbr 3924  cima 4537  Fun wfun 5112  ontowfo 5116  cfv 5118  1st c1st 6029  2nd c2nd 6030  Qcnq 7081   <Q cltq 7086  Pcnp 7092  <P cltp 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-qs 6428  df-ni 7105  df-nqqs 7149  df-inp 7267  df-iltp 7271
This theorem is referenced by:  suplocexpr  7526
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